Consider globally complete intersections in $\mathbb{P}^n$, of codimension $k$, of some fixed multi-degree $(d_1,\dots,d_k)$.

1. Is there some nice (i.e. "explicit") parameter space for them? (even if all the $d_i$'s are distinct one cannot take just the product of linear systems :) Or, does the corresponding locus in the Hilbert scheme admit some nice/explicit description? (a relevant question on this locus in Hilbert scheme)

2. Consider the discriminant in the parameter space of complete intersections, i.e. the locus parametrizing singular c.i.'s. I need standard results: it is an irreducible, reduced hypersurface, singular in codimension one etc. (I guess many of these properties do not depend on the particular choice of parameter space.) Where can I read about this? Somehow I did not find this in Gel'fand-Kapranov-Zelevinsky. And the book of Looijenga treats the local case only.

a somewhat related question

Consider globally complete intersections in $\mathbb{P}^n$, of codimension $k$, of some fixed multi-degree $(d_1,\dots,d_k)$.

1. Is there some nice (i.e. "explicit") parameter space for them? (even if all the $d_i$'s are distinct one cannot take just the product of linear systems :) Or, does the corresponding locus in the Hilbert scheme admit some nice/explicit description? (a relevant question on this locus in Hilbert scheme)

2. Consider the discriminant in the parameter space of complete intersections, i.e. the locus parametrizing singular c.i.'s. I need standard results: it is an irreducible, reduced hypersurface, singular in codimension one etc. (I guess many of these properties do not depend on the particular choice of parameter space.) Where can I read about this? Somehow I did not find this in Gel'fand-Kapranov-Zelevinsky. And the book of Looijenga treats the local case only.

a somewhat related question